The answer to your question can be found in "The Naming of Cats." It is implied that having a name that's particular enables a cat to stand proudly with a "perpendicular" tail. This cat will be able to behave with aplomb, "[spreading] out his whiskers, or [cherishing] his pride."
To showcase his point, the speaker provides a number of examples of these "particular" names. The three most memorable of these names, in my opinion, are Munkustrap, Bombalurina, and Jellylorum. The important point about these names is that they are unique. Being given one of these "particular" names makes a cat stand out, be noticed, and be important.
It is important to note that Eliot refers to three different names that cats should have, and that the "particular" name is the second of these. The first is a name "that the family use daily" and is assigned to them at birth. This name is not necessarily unique, and may for example be "Peter, Augustus, Alonzo, or James." The second "particular name" is unique to every cat, and is a name that enables him to stand out and be proud. Last in the list of a cat's names is a name that only the cat himself or herself knows, which is secret and "will never [be confessed]."
Eliot specifies that a cat's "particular" name needs to be dignified in order to serve its purpose. The cat gains dignity from his or her dignified name.